A new family of maximum linear symmetric rank-distance codes

Let $\mathscr{S}_n(q)$ denote the set of symmetric bilinear forms over an $n$-dimensional $\mathbb{F}_q$-vector space. A subset $\mathcal{C}$ of $\mathscr{S}_n(q)$ is called a $d$-code if the rank of $A-B$ is larger than or equal to $d$ for any distinct $A$ and $B$ in $\mathcal{C}$. If $\mathcal{C}$ is further closed under matrix addition, then $|\mathcal{C}|$ is sharply upper bounded by $q^{n(n-d+2)/2}$ if $n-d$ is even and $q^{(n+1)(n-d+1)/2}$ if $n-d$ is odd. Additive codes meeting these upper bounds are called maximum. There are very few known constructions of them. In this paper, we obtain a new family of maximum $\mathbb{F}_q$-linear $(n-2)$-codes in $\mathscr{S}_n(q)$ for $n=6,8$ and $10$ which are not equivalent to any known constructions. Furthermore, we completely determine the equivalence between distinct members in this new family.